{"id":{"repo_id":"etsu","oai_identifier":"oai:dc.etsu.edu:etd-1039"},"canonical_url":"https://search.dev.ndltd.org/etd/etsu/oai:dc.etsu.edu:etd-1039","repository":{"repo_id":"etsu","name":"East Tennessee State University","base_url":"https://dc.etsu.edu/do/oai/"},"display":{"title":"A Characterization of Large (<em>t,r</em>)-Regular Graphs.","abstract":"<p>A graph <em>G</em> is a (<em>t</em>,<em>r</em>)-regular graph if every collection of <em>t</em> independent vertices is collectively adjacent to exactly <em>r</em> vertices. In this thesis, we will present a complete characterization of (<em>t</em>,<em>r</em>)-regular graphs of order <em>n</em> if <em>n</em> is sufficiently large. Furthermore, we will show that all graphs of this type are isomorphic to <em>K<sub>s</sub></em> + <em>mK<sub>p</sub></em> where <em>t</em>(<em>p</em> - 1) + <em>s</em> = <em>r</em>.</p>","abstract_html":"&lt;p&gt;A graph &lt;em&gt;G&lt;/em&gt; is a (&lt;em&gt;t&lt;/em&gt;,&lt;em&gt;r&lt;/em&gt;)-regular graph if every collection of &lt;em&gt;t&lt;/em&gt; independent vertices is collectively adjacent to exactly &lt;em&gt;r&lt;/em&gt; vertices. In this thesis, we will present a complete characterization of (&lt;em&gt;t&lt;/em&gt;,&lt;em&gt;r&lt;/em&gt;)-regular graphs of order &lt;em&gt;n&lt;/em&gt; if &lt;em&gt;n&lt;/em&gt; is sufficiently large. Furthermore, we will show that all graphs of this type are isomorphic to &lt;em&gt;K&lt;sub&gt;s&lt;/sub&gt;&lt;/em&gt; + &lt;em&gt;mK&lt;sub&gt;p&lt;/sub&gt;&lt;/em&gt; where &lt;em&gt;t&lt;/em&gt;(&lt;em&gt;p&lt;/em&gt; - 1) + &lt;em&gt;s&lt;/em&gt; = &lt;em&gt;r&lt;/em&gt;.&lt;/p&gt;","abstract_has_math":false,"creators":["Gosnell, Shannon Leah"],"institution":"East Tennessee State University","degree_name":"MS (Master of Science)","degree_level":"Thesis - restricted","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2000,"date_issued":"2000-05-01T07:00:00Z","date_published":"2000-05-01T07:00:00Z","updated_at":"2026-07-24T02:18:50Z","subjects":["graph theory","regular","(t r)-regular","r-regular","Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":["Copyright by the authors."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.etsu.edu/etd/7","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Gosnell, Shannon Leah"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2000-01-01T08:00:00Z"]},{"key":"dc:date.issued","label":"Date","values":["2000-05-01T07:00:00Z"]},{"key":"dc:publisher","label":"Institution","values":["East Tennessee State University"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis - restricted"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS (Master of Science)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["graph theory","regular","(t r)-regular","r-regular","Mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Copyright by the authors."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dc.etsu.edu/context/etd/article/1039/viewcontent/Thesis.pdf","https://dc.etsu.edu/etd/7"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["<p>A graph <em>G</em> is a (<em>t</em>,<em>r</em>)-regular graph if every collection of <em>t</em> independent vertices is collectively adjacent to exactly <em>r</em> vertices. In this thesis, we will present a complete characterization of (<em>t</em>,<em>r</em>)-regular graphs of order <em>n</em> if <em>n</em> is sufficiently large. Furthermore, we will show that all graphs of this type are isomorphic to <em>K<sub>s</sub></em> + <em>mK<sub>p</sub></em> where <em>t</em>(<em>p</em> - 1) + <em>s</em> = <em>r</em>.</p>"]},{"key":"dc:title","label":"Title","values":["A Characterization of Large (<em>t,r</em>)-Regular Graphs."]}]}],"canonical_facts":{"dc:creator":["Gosnell, Shannon Leah"],"dc:date.available":["2000-01-01T08:00:00Z"],"dc:date.issued":["2000-05-01T07:00:00Z"],"dc:description.abstract":["<p>A graph <em>G</em> is a (<em>t</em>,<em>r</em>)-regular graph if every collection of <em>t</em> independent vertices is collectively adjacent to exactly <em>r</em> vertices. In this thesis, we will present a complete characterization of (<em>t</em>,<em>r</em>)-regular graphs of order <em>n</em> if <em>n</em> is sufficiently large. Furthermore, we will show that all graphs of this type are isomorphic to <em>K<sub>s</sub></em> + <em>mK<sub>p</sub></em> where <em>t</em>(<em>p</em> - 1) + <em>s</em> = <em>r</em>.</p>"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/1039/viewcontent/Thesis.pdf","https://dc.etsu.edu/etd/7"],"dc:publisher":["East Tennessee State University"],"dc:rights":["Copyright by the authors."],"dc:subject":["graph theory","regular","(t r)-regular","r-regular","Mathematics","Physical Sciences and Mathematics"],"dc:title":["A Characterization of Large (<em>t,r</em>)-Regular Graphs."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - restricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:18:50Z"}